The Tale of OpenGL vs. Direct3D
Many of the answers here are really, really good. But the OpenGL and D3D issue should probably be addressed. And that requires... a history lesson.
And before we begin, I know far more about OpenGL than I do about Direct3D. I've never written a line of D3D code in my life, and I've written tutorials on OpenGL. So what I'm about to say isn't a question of bias. It is simply a matter of history.
Birth of Conflict
One day, sometime in the early 90's, Microsoft looked around. They saw the SNES and Sega Genesis being awesome, running lots of action games and such. And they saw DOS. Developers coded DOS games like console games: direct to the metal. Unlike consoles however, where a developer who made an SNES game knew what hardware the user would have, DOS developers had to write for multiple possible configurations. And this is rather harder than it sounds.